Heat the remaining 2 teaspoons oil in a large, ovenproof nonstick skillet over medium heat. Cook the chicken until browned on both sides, about 2 minutes per side. Move the chicken to the center so all pieces are touching. Spread with the cheese mixture, sprinkle with ham, then top with the breadcrumb mixture.

Bake until the chicken is no longer pink in the center and an instant-read thermometer registers 165°F, 5 to 7 minutes.